Method and apparatus for producing a porosity log of a subsurface formation corrected for detector standoff

ABSTRACT

A borehole logging tool is lowered into a borehole traversing a subsurface formation and a neutron detector measures the die-away of nuclear radiation in the formation. A model of the die-away of nuclear radiation is produced using exponential terms varying as the sum of borehole, formation and thermal neutron background components. Exponentially weighted moments of both the die-away measurements and die-away model are determined and equated. The equated moments are solved for the ratio of the borehole to formation amplitude components of the measurements. The formation decay constant is determined from at least the formation and thermal neutron background terms of the weighted measurement and model moments. The determined borehole to formation amplitude ratio is used to correct the determined formation decay constant for the effects of detector standoff from the borehole wall. A porosity log of the formation is produced which is corrected for detector standoff from the borehole wall as a function of the standoff corrected formation decay constant calibrated in borehole models of known porosities.

BACKGROUND OF THE INVENTION

This invention relates to a borehole logging method for measuring thedie-away of nuclear radiation of a subsurface formation and forcorrecting signals representative of the decay constants and amplitudesof the measured radiation for the effects of detector standoff from theborehole wall of the formation where such signals may vary with time,distance, or any other independent variable. A porosity versus depth logis then produced for the subsurface formation as a function of thestandoff corrected formation decay constant derived from calibratedmeasurements in borehole models of known porosities and conditions ofdetector standoff.

A borehole logging tool is conventionally used to measure the die-awayof nuclear radiation in the formations surrounding a borehole. Inneutron porosity logging, for example, a neutron source is utilized forbombarding the formations with fast neutrons as the logging tool isadvanced through the borehole. In the course of moderation, the fastneutrons reach the epithermal range and thence are further moderateduntil they reach the thermal neutron range. The populations of neutronsat the various energy levels die-away with time following primaryirradiation and thus offer means of characterizing the formations. Therate of die-away of the epithermal neutron population gives aquantitative indication of the amount of hydrogenous material present inthe formations which in turn is indicative of the porosities of theformations. Examples of both methods and apparatus for carrying outepithermal die-away porosity logging are set forth in U.S. Pat. No.4,097,737 to W. R. Mills, Jr., and U.S. Pat. Nos. 4,556,793 and4,590,370 to L. S. Allen and W. R. Mills, Jr.

An article entitled "Improved Methods of Signal Processing For PulsedNeutron Capture Logging", SPWLA Twenty Second Annual Logging Symposium,June 23-26, 1981 by R. Randall and E. C. Hopkinson discusses a method ofpulsed neutron capture logging to differentiate oil, gas and salineformation water environments through casing. The method appliesstatistical averaging to a single exponential die-away term in thelogged data to determine thermal neutron decay rate. In a still furtherreference, U.S. Pat. No. 4,600,838 to D. K. Steinman and L. A. Jacobson,there is described a method of thermal neutron die-away logging forovercoming excessive statistical fluctuations in the logged data,particularly in strongly absorbing formations. This method involves thedetermination of zero and first order moments of time during first andsecond discrete sequences of time gates respectively and the taking ofthe ratio of such moments to obtain a thermal neutron decay constant forthe formations surrounding the logged borehole. In yet furtherreferences, Smith, Jr. U.S. Pat. No. 4,625,110, and U.S. Pat. No.4,638,161 to Smith, Jr. and Verbout, there is described a porositydetermination utilizing a two-exponential model approach to epithermalneutron die-away. Finally, Loomis U.S. Pat. No. 4,972,082 discloses useof a modulated exponential function to provide correction of anepithermal neutron die-away measurement for the situation where thelogging tool is not in good contact with the borehole wall.

While the foregoing described methods and systems have been utilized inthe production of well logs for characterizing the die-away of nuclearradiation of subsurface formations, there is still need for a verysensitive method of improving on the characterizations provided byporosity logs when the neutron detector of the logging tool is not incontact with the formation (i.e., separation or standoff of the loggingtool detector from the borehole wall). Hereinafter, such separationswill be referred to as standoff, although they can arise from thelogging tool pulling away from the surface of the borehole wall or fromthe borehole enlarging away from the logging tool. In such cases, theformation decay constant determined from the die-away of nuclearradiation measurements will be in error by the degree of effect suchstandoff has on the measurement. It is therefore an object of thepresent invention to provide for a porosity log generated from astandoff corrected formation decay constant such that the formation maybe accurately characterized as to porosity.

SUMMARY OF THE INVENTION

The present invention is directed to a method for measuring the die-awayof nuclear radiation of a subsurface formation surrounding a boreholeand for correcting such measurements for the effect of logging tooldetector standoff from the borehole wall of the formation so that anaccurate formation porosity log may be produced.

A logging tool is lowered into a borehole traversing a subsurfaceformation of interest and measurements are obtained of the die-away ofnuclear radiation in the formation. Signals are generated representativeof the intensity of the measurements. A model of such signals isgenerated having three exponential terms varying as the sum of theborehole, formation and thermal neutron background effects on suchmeasurements. This is a complete representation of the principalcomponents of the die-away process and provides a very sensitiveparameter for correcting measured lifetimes for logging tool standoff.Three exponentially weighted signal moments and three exponentiallyweighted model moments are determined. These moments are equated andsolved for the ratio of the borehole amplitude to the formationamplitude components of the signals. An initial formation decay constantis determined from a solution of the moment equations which involves atleast the formation and thermal neutron background terms of theexponentially weighted signal and model moments. This formation decayconstant is corrected for standoff of the detector from the boreholewall in accordance with a function relating such decay constant to theratio of borehole to formation amplitudes. This function is derived fromintensity logging measurements taken in borehole models at severalporosities and conditions of logging tool separation from the formation.A porosity versus depth log is produced as a function of the standoffcorrected formation decay constant derived from calibration measurementsin the borehole models at known porosities and conditions of detectorstandoff from the borehole wall.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 illustrates a borehole logging system for making die-awaymeasurements of nuclear radiation from a subsurface formationsurrounding a borehole and for producing a porosity versus depth log ofsuch formation.

FIG. 2 illustrates a specific example of epithermal neutron die-awaymeasurements as might be made by the borehole logging system of FIG. 1.

FIG. 3 illustrates the downhole and uphole electronics units of theborehole logging system of FIG. 1.

FIGS. 4 & 5 are flow charts representing the steps carried out inaccordance with the method of the present invention for correcting thedie-away of nuclear radiation measurements of FIG. 2 for the effect ofdetector standoff from the formation.

DESCRIPTION OF THE PREFERRED EMBODIMENT

Referring to FIG. 1, there is illustrated a borehole logging systemuseful in logging a subsurface formation for porosity in accordance withthe present invention.

A borehole tool 10, supported by cable 11, comprises a high energypulsed neutron source 12 and an articulated radiation detector pad 13. Ahigh-voltage power supply 14 is provided for the source 12 and adownhole electronics unit 15 is provided with circuits for utilizationand modification of signals from radiation detector pad 13. Alsoincluded are circuits for the control of the high voltage power supply14. A backup arm 16 with attached pad 18 extending from the logging tool10 may be included to force the logging tool 10 to follow the averagecontour of the borehole wall 24. Cable 11 extends to a surfaceelectronics unit 17 where the signals from the radiation detector pad 13are processed and recorded, along with an indication of depth, as aporosity log of the subsurface formation.

Particularly troublesome, however, to such a radiation measurement isthe presence of rugosity in the borehole wall. Any borehole enlargementof depth greater than about one-eighth inch can seriously affect theformation radiation measurement.

To help eliminate the troublesome borehole rugosity effects on theformation radiation measurement, the radiation detector should be housedin a pad 13 which is small enough to track short borehole enlargements.Most such enlargements can be tracked by a radiation detector pad thatis on the order of one foot in length. The radiation detector pad 13includes at least one epithermal neutron detector 19 which is positionedagainst one side of pad 13, and a shielding material (i.e., a moderatingmaterial with or without an appropriate absorbing material) (not shown)which fills the remaining portion of the pad. Such a detector isillustrated in Givens U.S. Pat. No. 4,581,532 and in Allen U.S. Pat. No.4,692,617, the teachings of which are incorporated herein by reference.The arm 22, shown in FIG. 1, azimuthally orients the pad so that theside of the pad against which the neutron detector is positioned isfirmly pressed against the surface of the borehole wall. In this manner,the focusing of the directional sensitivity of the assembly consistingof the neutron detector and the shielding material on the formation ismaximized, while the directional sensitivity of such assembly radiationfrom the borehole fluid is minimized by the moderation and/or absorptionof such borehole fluid radiation by the shielding material. Articulatingarm 22 is shown in FIG. 1 for purposes only as one example of a meansfor positioning the radiation detector against the surface of theborehole wall as is illustrated in the aforementioned Allen U.S. Pat.No. 4,692,617. Other positioning means may also be acceptable such as bylocating the radiation detector in the main body of the logging tool andpressing the tool or that portion of the tool housing the radiationdetector against the surface of the borehole wall as shown in theaforementioned Givens U.S. Pat. No. 4,581,532 or in the aforementionedLoomis U.S. Pat. No. 4,972,082, the teaching of which is alsoincorporated herein by reference.

Nevertheless, there is still a need for improving on thecharacterization of the subsurface formation as to porosity in thosesituations where there is standoff of the neutron detector from theformation, whether due to the pad 13 not being completely in directcontact with the borehole wall or whether due to the entire tool 10 notbeing in complete contact with the borehole should the articulated pad13 not be employed and the detector be alternatively housed in the mainbody of the tool 10. Such improvement, in accordance with a specificfeature of the present invention, involves the correcting of theformation decay constant used in producing the porosity log for suchdetector standoff in accordance with a function relating formation decayconstant to the ratio of borehole to formation components of die awayamplitude, particularly where the measured die-away is comprised of aplurality of exponential terms, such as represented by the die-awaycurve illustrated in FIG. 2. For purposes of illustration, the curve inFIG. 2 is assumed to consist of the sum of two epithermal neutrondie-away components and one thermal neutron die-away component. A firstexponential die-away component A₁ e⁻μ 1^(t) from 0 to t₁ microsecondsrepresents predominantly the rate of die-away of epithermal neutronsattributable to borehole effects. A second exponential die-awaycomponent A₂ e⁻μ 2 ^(t) from t₁ to t₂ microseconds representspredominantly the rate of die-away of epithermal neutrons attributableto formation effects. A third die-away component A₃ e⁻μ 3^(t) from t₂ tot_(f) microseconds represents predominantly the rate of die-awayattributable to thermal neutron effects. Each of these components canfrequently be observed as a straight line when total intensity isplotted on a semilogarithmic scale during the time domain when theeffect is dominant. For a more detailed description of such a pluralityof die-away characteristics, as well as a description of a boreholelogging apparatus 10, including surface equipment, for making thedie-away data measurements, reference may be had to the aforementionedU.S. Pat. Nos. 4,097,737; 4,556,793; and 4,590,370, the teachings ofwhich are incorporated herein by reference.

Referring now to FIG. 3 there is shown the downhole electronics 15 andsurface electronics 17 useful for processing the die-away signals fromthe nuclear detector and producing the desired porosity log correctedfor standoff. The die away signal from the neutron detector within pad13 of FIG. 1 is applied through amplifier 25 and pulse heightdiscriminator 26 to a time analyzer 28.

Time analyzer 28 stores the incoming die-away signal as a multichanneltime sequence of counts representing the die-away spectrum. Preferablythis is a sequence of contiguous channels each one microsecond in width,but other recording modes could be used. The die-away signal isaccumulated over a time frame of approximately one second andcorresponds to approximately 5000 firings of the neutron source. Thespectrum thus accumulated is then passed to uphole electronics 17 forprocessing.

The uphole electronics 17 is controlled by a master digital computer(Hewlitt Packard Model 1000 for example). A first signal processor(Stage I) 30 solves equations which relate time moments calculated forthe accumulated data spectrum with time moments calculated for themathematical model of the die-away process. This solution produces theparameters A_(i) and μ_(i) previously described.

The parameters A_(i) and μ_(i) are then passed to a second signalprocessor (Stage II) 31 where the formation die-away parameter μ₂ iscorrected for standoff effects by the ratio R of the amplitude of itsborehole die-away component A₁, as described above, to the amplitude ofits formation component A₂. Thereafter a standoff corrected formationdecay constant is produced and applied to function former unit 32 whichutilizes calibration measurements taken in borehole models at knownporosities and conditions of detector standoff to produce the desiredporosity versus depth recording 33, or log.

A detailed explanation of the operation of the uphole electronic signalprocessing will now be made in conjunction with the flow charts of FIGS.4 and 5. The characteristic parameter requiring correction is thelifetime (or decay constant) of the formation and the characteristicparameter which can be used for correction is the ratio of the amplitudeof the borehole component to the amplitude of the formation component ofthe die-away measurement. This amplitude ratio can be used to correctthe measured lifetime for situations where the detector is not in directcontact with the borehole wall because the effect of the standoffcorrelates strongly with the amplitude ratio when the time dependence ofthe borehole measurement is modeled as the sum of three exponentialfunctions: a borehole component, a formation component and a thermalneutron "background" component, as shown in steps 410-480 of FIG. 4. Themeasured epithermal neutron signal f(t) can be expressed as:

    f(t)=A.sub.1 e.sup.-μ 1.sup.t +A.sub.2 e.sup.-μ 2.sup.t +A.sub.3 e.sup.-μ 3.sup.t                                       (1)

where the capital letters A₁ -A₃ denote amplitudes and the Greek lowercase letters μ₁ -μ₃ denote decay constants of the three components. Inthis representation the letters A₁ and μ₁ apply to the boreholecomponent, the letters A₂ and μ₂ apply to the formation component, andthe letters A₃ and μ₃ apply to the thermal neutron background component.

If the radiation detector is strongly focused on the formation byappropriate shielding and by tool design and positioning, then decayconstant μ₁ can be treated as a constant parameter determined bycalibration of the measurement in suitable borehole models of knownphysical characteristics. Decay constant μ₃, although varying withformation and borehole properties, can be assumed a constant averagevalue because of the small effect of the thermal neutron backgroundcomponent on the epithermal neutron measurement (A₃ <<A₁ or A₂). Thisleaves the decay constant μ₂ and the amplitudes A₁, A₂ and A₃ to bedetermined from the borehole measurement (steps 410-420). Threeparameters can be determined from the calculation of three weightedmoments (step 430) of the epithermal neutron flux; these are A₁ /A₂, A₃/A₂ and μ₂. It is not necessary to solve for the absolute magnitudes ofA₁, A₂ or A₃.

If exponential weighting is used to suppress the effect of later datachannels with increasingly large statistical uncertainties, the expectedvalues of the three model moments of time are: ##EQU1## where

    w.sub.ji =e.sup.-z j.sup.t i,                              (3)

j=1, 2, 3,

and the sums are over the appropriate time intervals where theexponential model is valid. Reasonable choices for the z_(j) range fromabout 0.01 inverse microsecond to about 0.5 inverse microsecond. Threeintensity moments are expressed: ##EQU2## j=1, 2, 3, where N_(i) is theintensity of the recorded data.

The three exponentially weighted model and intensity moments are equatedand the resulting equations solved for A₁ /A₂, A_(3/A) ₂ and μ₂.

The exponentially varying intensity data are measured as a continuousfunction of time or in discrete time intervals. Firstly, with respect tothe continuous case, the intensity data are modeled as follows: ##EQU3##where μ'_(k) are the decay constants for the continuous case and A'_(k)are the amplitudes. The expected value of weighted moments of time withexponential weighting are determined from the model as follows: ##EQU4##for each z_(j).

The weighted moments of data are then computed as: ##EQU5## where y(t)represents the continuous data.

Next the corresponding model and intensity moments are equated:

    E[t.sub.j (t.sub.a,t.sub.b)]=t.sub.j (t.sub.a,t.sub.b)     (8)

for all three weighting functions yielding three transcendentalequations. The three equations are solved simultaneously for estimatesof the parameters A'₁ /A'₂, A'₃ /A'₂ and the formation component decayconstant μ₂ ' with the borehole component decay constant μ'₁ and thethermal neutron component decay constant μ'₃ being known (step 440). Itis not necessary to determine A'₂. The starting time t_(a) and endingtime t_(b) are illustrated in FIG. 2. For the continuous case A'_(k) hasunits of counts per unit interval and μ'_(k) has units of inverse unitinterval.

Secondly, with respect to the discrete case, the intensity data aremodeled as follows: ##EQU6## where n=1, . . . , N. The decay constantsare μ"_(k) for the discrete case and the amplitudes are A"_(k). Theexpected value of weighted moments of time interval with exponentialweighting are determined from the model as follows: ##EQU7## for eachZ'_(j) (different from Z_(j) only by conversion to discrete timechannels). The weighted moments of intensity are then computed asfollows: ##EQU8## where Y(n) represents the discrete data. Next thecorresponding model and intensity moments are equated:

    E[n.sub.j (n.sub.a,n.sub.b)]=n.sub.j (n.sub.a,n.sub.b)     (12)

yielding three transcendental equations which are solved simultaneouslyfor estimates of the three parameters A"₁ /A"₂, A"₃ /A"₂ and theformation component decay constant μ"₂ with the borehole component decayconstant μ"₁ and the thermal neutron component decay constant μ"₃ beingknown. For the discrete case A"_(k) has units of counts per unit timeand μ"_(k) has units of inverse unit time.

Reasonable choices for the Z'_(j) are 1/64, 1/32 and 1/16 when the timechannels are each 1 μs in duration.

After solving the resulting transcendental equation for the modelparameters by an iterative technique or Newton's method, a decision ismade on the need for employing a double exponential model. The decisioncan be based on the decay constant μ₂ (i.e., μ'₂ for the continuous caseor μ"₂ for the discrete case) or the amplitude ratio A₁ /A₂ (hereinaftertermed R). Illustrated in FIG. 4 is the use solely of μ₂ as the decisionparameter (step 450). The standoff-corrected decay constant μ₂ * is thencalculated from the empirical equation for both the continuous anddiscrete cases (step 460):

    μ.sub.2 *=f.sub.1 (μ.sub.2,R)                        (13)

A useful form for the function of Equation (14) is:

    (1/x*)=(1/x) (1+a(R-R.sub.o)), or                          (14)

    (1/x*)=(1/x) (1+b(1/x))                                    (15)

where

    R.sub.o =c+dx+ex.sup.2,                                    (16)

a-e are empirical constants and x is either μ₂ ' for the continuous caseor μ₂ " for the discrete case.

Having obtained a corrected value for μ₂, conversion to a porosity Pversus depth log is done by a second empirical equation (step 470):

    P=f.sub.2 (μ2*)                                         (17)

A useful form is:

    P=g+hx+mx.sup.2 +px.sup.3                                  (18)

where x again stands for μ₂ ' or μ₂ " and g, h, m and p are empiricalconstants. The empirical constants a-e, g, h, m and p are obtained fromlogging tool measurements made in borehole models under knownconditions. It is obvious that the amplitude ratio R can be calibratedin terms of detector standoff and that this apparent standoff can be anoutput value if desired.

Clearly, to obtain two accurate empirical equations for equations (13)and (17), it is necessary to have measurements made in borehole modelsat several porosities for a variety of separation conditions. Apractical minimum is probably four porosities and four differentconditions of detector formation separation. Many more combinationswould be helpful. The experimental measurements can obviously beaugmented by accurate theoretical calculations, and these should be usedwhen practical. It should be apparent to those skilled in the art thatsuch empirical functions can have many acceptable forms other than thoseset forth above in equations (14), (15), (16) and (18). The best form isusually selected on some goodness of fit criterion.

One additional step, alluded to earlier, should be taken to enhanceresults. When standoff is large or when porosity is high the number ofcounts recorded in a time analyzer is reduced and the three-exponentialmodel will produce results having larger statistical fluctuations thandesired. In this case the analysis for the decay constant μ₂ is switchedto a two-exponential model, as shown in steps 500-530 of FIG. 5,representing only the formation and the thermal neutron background. Thegeneral equation for f(t) symbolically representing both the continuousand discrete cases is then ##EQU9##

The stand-off corrected decay constant is then calculated for either thecontinuous or discrete cases (step 510):

    μ.sub.2 *=f.sub.3 (μ.sub.2,R)                        (21)

Conversion to the porosity versus depth log is then derived by (step520):

    P=f.sub.4 (μ.sub.2 *)                                   (22)

This use of a two-exponential model may require that the analysis bestarted a few channels later in time, but it still produces superiorresults because the three-exponential model has become so dominated bythe borehole component that measurement sensitivity to the desiredformation component is reduced. The switch from three exponentials totwo exponentials can be based on statistical considerations, but inpractice it may be desirable to make the change at a porositycorresponding to the upper limit of very well-consolidated formations. Aporosity of approximately 20 percent should avoid serious boreholerugosity problems at the point of change. Note that the amplitude ratioR from the three-exponential model is always used to determine theeffect of detector formation separation, even when the two-exponentialmodel is used to determine μ₂. It will normally be desirable to displayon the porosity log the magnitude of the porosity correction (inporosity units) as well as the corrected porosity. The amplitude ratiomay also be displayed directly as an indicator of rugosity or it can becalibrated and displayed as an apparent standoff.

It should be clear to those skilled in the art that the data momentcalculations used in the three-exponential model can be used in thetwo-exponential model without recalculation and that the well-knownequation for the sum of a finite geometric series should be used tosimplify certain sums in the model representations.

Having now described the present invention in conjunction with apreferred embodiment, it is to be understood that various modificationsand alterations may be apparent from the spirit and scope of theinvention as set forth in the appended claims. For example, if amplitudeA₃ were made to be very small (i.e., A₃ ≈0) through use of a thickthermal neutron shield, then only two weighted moments are needed tosolve for decay constant μ₂ and amplitude ratio A₁ /A₂ provided decayconstant μ₁ is fixed. If μ₁ is not fixed, then three weighted momentsmust be determined to obtain μ₁,μ₂ and A₁ /A₂.

Further, a modulated exponential form may be used to represent the data.One of the simplest forms is: ##EQU10## where the α_(i) are constants.The α_(i) will normally be determined from measurements in boreholemodels. This avoids the calculation of additional moments in thereal-time data analysis algorithm. Such modulated exponential forms mayfit the somewhat scattered data that must normally be worked withsatisfactorily. However, they generally result in expressions involvingthe gamma function and are thus not as convenient to use as expressionsinvolving the standard exponential function.

Still further, the calibrated output could be hydrogen index instead ofporosity. The lifetime measurement is most generally calibrated in termsof hydrogen concentration and could be presented this way. It is assumedthat the pore space is completely saturated with fresh water in order topresent a porosity curve. False (i.e., low) porosity values are recordedin low-pressure gas zones and in partially-saturated zones which areabove the water table.

We claim:
 1. A method for converting signals representing the die-awayof nuclear radiation in a subsurface formation surrounding a boreholeinto a log representing porosity versus depth within said subsurfaceformation corrected for the effect of detector standoff from theborehole wall,a) lowering a logging tool having a neutron source and aneutron detector into said borehole, b) irradiating said subsurfaceformation with neutrons from said neutron source as said logging tool istraversed along said subsurface formation, c) recording die-away signalsrepresenting the die-away of nuclear radiation in said subsurfaceformation as detected by said neutron detector, d) producing intensitysignals representing the variations in intensity of said die-awaysignals, e) producing a model of the die-away of nuclear radiation insaid subsurface formation having terms varying exponentially in responseto borehole, formation and background effects on the die-away of nuclearradiation as detected by said detector, f) producing weighted moments ofsaid model and of said intensity signals, g) producing a ratio of theborehole amplitude to formation amplitude components of said intensitysignals derived from said weighted moments, h) producing an initialformation decay constant signal from formation and thermal neutronbackground terms of said model and said intensity signals, i) producinga standoff corrected formation decay constant signal in accordance witha function of said ratio and said initial formation decay constantsignal derived from die-away signals recorded in borehole models atknown porosities and conditions of detector standoff from the boreholewall, and j) producing a log of porosity versus depth within saidsubsurface formation corrected for detector standoff from the boreholewall as a function of said standoff corrected formation decay constantsignal derived from calibrated measurements in borehole models at knownporosities and conditions of detector standoff from the borehole wall.2. The method of claim 1 wherein said model of the die-away of nuclearradiation is produced from a sum of exponential terms as follows:

    f(t)=A.sub.1 e.sup.-μ 1.sup.t +A.sub.2 e.sup.-μ 2.sup.t +A.sub.3 e.sup.-μ 3.sup.t

where A₁ =borehole amplitude A₂ =formation amplitude A₃ =backgroundamplitude, μ₁ =borehole decay constant, μ₂ =formation decay constant,and μ₃ =background decay constant.
 3. The method of claim 1 wherein:a)said weighted moments of said model are produced as follows:

    E[t.sub.j]=Σt.sub.i w.sub.ji f(t.sub.i)/Σw.sub.jji f(t.sub.i),

wherew_(ji) =e^(-z) j^(t) i, z_(j) =weighting function decay constantj=1,2,3, and b) said weighted moments of said intensity signals areproduced as follows: ##EQU11## where N_(i) =intensity of the recordeddata.
 4. A method for converting signals representing the die-away ofnuclear radiation in a subsurface formation surrounding a borehole intoa log representing porosity versus depth within said subsurfaceformation corrected for the effect of detector standoff from theborehole wall,a) lowering a logging tool having a neutron source and aneutron detector into said borehole, b) irradiating said subsurfaceformation with neutrons from said neutron source as said logging tool istraversed along said subsurface formation, c) recording die-away signalsrepresenting the die-away of nuclear radiation in said subsurfaceformation as detected by said neutron detector, d) producing intensitysignals representing the variations in intensity of said die-awaysignals, e) producing a model of the die-away of nuclear radiation insaid subsurface formation having three exponential terms varying inresponse to borehole, formation and thermal neutron background effectson said die-away signals as follows: ##EQU12## where μ.sub. = boreholedecay constant,μ₂ =formation decay constant, μ₃ =thermal neutronbackground decay constant A₁ =borehole amplitude, A₂ =formationamplitude, A₃ =thermal neutron background amplitude, t=time, f)determining the expected value of weighted model moments withexponential weighting in accordance with the three-exponential model ofstep e) as follows: ##EQU13## where z_(j) =weighting function decayconstantt_(a) =starting time, t_(b) =ending time, g) determining weighedmoments of said intensity signals with exponential weighting inaccordance with the three-exponential model of step d) as follows:##EQU14## where y(t)=intensity h) equating said weighted model momentswith said weighted intensity moments;

    E[t.sub.j (t.sub.a,t.sub.b)]=t.sub.j (t.sub.a,t.sub.b)

i) utilizing said equated weighted model and intensity signal momentsfrom step h) to determine a ratio signal R of the borehole amplitude A₁to the formation amplitude A₂ components of said intensity signal:

    R=A.sub.1 /A.sub.2,

j) determining an initial formation decay constant μ₂ utilizing at leastthe two-exponential terms of said equated weighted model and intensitysignal moments from step h) attributable to the formation and thermalneutron background effects, k) determining a standoff correctedformation decay constant μ₂ * for detector standoff from the boreholewall in accordance with a function f(μ₂,R), derived from calibrationmeasurements taken in borehole models at known porosities and conditionsof detector standoff from the formation:

    μ.sub.  *=f(μ.sub.2,R), and

l) producing a porosity versus depth log of said subsurface formationcorrected for detector standoff from the borehole wall as a function ofsaid standoff corrected decay constant μ₂ * derived from calibratedmeasurements in said borehole models at said known porosities andconditions of detector standoff from the borehole wall.
 5. The method ofclaim 4 wherein step j) of determining the initial formation decayconstant μ₂ is carried out utilizing all three exponential terms of saidequated weighted model and intensity moments attributable to borehole,formation and thermal neutron background effects.
 6. A method forconverting signals representing the die-away of nuclear radiation in asubsurface formation surrounding a borehole into a log representingporosity versus depth within said subsurface formation corrected for theeffect of detector standoff from the borehole wall,a) lowering a loggingtool having a neutron source and a neutron detector into said borehole,b) irradiating said subsurface formation with neutrons from said neutronsource as said logging tool is traversed along said subsurfaceformation, c) recording die-away signals representing the die-away ofnuclear radiation in said subsurface formation as detected by saidneutron detector, d) producing intensity signals representing thevariations in intensity of said die-away signals, e) producing a modelof the die-away of nuclear radiation in said subsurface formation havingthree exponential terms varying in response to borehole, formation andthermal neutron background effects on said measurements as follows:##EQU15## where μ.sub. = borehole decay constant,μ.sub. = formationdecay constant, μ₃ =thermal neutron background decay constant A₁=borehole amplitude, A₂ =formation amplitude, A₃ =thermal neutronbackground amplitude, and n=1, 2, . . . , N (discrete intervals of time)f) determining the expected value of weighted model moments withexponential weighting in accordance with the three-exponential model ofstep e) as follows: ##EQU16## where z_(j) =weighting function decayconstantn_(a) =discrete interval starting time, n_(b) =discrete intervalending time, g) determining weighted moments of said intensity signalswith exponential weighting in accordance with the three-exponentialmodel of step d) as follows: ##EQU17## where Y(n) represents discretedata, h) equating said weighted model moments with said weightedintensity moments:

    E[n.sub.j (n.sub.a,n.sub.b)]=n.sub.j (n.sub.a,n.sub.b),

i) utilizing said equated weighted model and intensity moments from stepe) to determine a ratio signal R of the borehole amplitude A₁ to theformation amplitude A₂ components of said intensity signal:

    R=A.sub.1 A.sub.2,

j) determining an initial formation decay constant μ₂ utilizing at leastthe two-exponential terms of said equated weighted model and intensitysignal moments from step h) attributable to the formation and thermalneutron background effects, and k) determining a standoff correctedformation decay constant μ₂ * for detector standoff from the boreholewall in accordance with a function f(μ₂,R), derived from calibrationmeasurements taken in borehole models at known porosities and conditionsof detector standoff from the formation:

    μ.sub.  *=f(μ.sub.2,R), and

l) producing a porosity versus depth log of said subsurface formationcorrected for detector standoff from the borehole wall as a function ofsaid standoff corrected decay constant μ₂ * derived from calibratedmeasurements in said borehole models at said known porosities andconditions of detector standoff from the borehole wall.
 7. The method ofclaim 6 wherein step j) of determining the initial formation decayconstant μ₂ is carried out utilizing all three exponential terms of saidequated weighted model and intensity signal moments attributable toborehole, formation and thermal neutron background effects.
 8. Apparatusfor converting die-away signals representing the die-away of nuclearradiation in a subsurface formation surrounding a borehole into a logrepresenting porosity versus depth within said formation corrected forthe effect of detector standoff from the borehole wall, comprising:a) aborehole logging tool, b) means for traversing said logging tool alongsaid borehole, c) a pulsed source of fast neutrons within said loggingtool for irradiating the formation surrounding a borehole with fastneutrons, d) at least one directionally sensitive radiation detectorassembly having a neutron detector for measuring the die-away ofsecondary radiation produced by the fast neutrons which returns to theborehole from the irradiated formation, e) means for positioning said atleast one radiation detector assembly so that it contacts the surface ofthe borehole wall and is oriented to position the neutron detector tomaximize directional sensitivity of the neutron detector to radiationfrom the formation at the point of contact of the at least one radiationdetector assembly with the borehole wall and to minimize directionalsensitivity to radiation from the borehole fluid, f) means for producingdie-away signals representative of the secondary radiation measured bysaid detector as said logging tool traverses said borehole, g) means forproducing an initial formation decay constant from at least formationand background components of said die-away signals, h) means forproducing a ratio of the borehole amplitude to formation amplitudecomponents of said die-away signals, i) first function former means forproducing a standoff corrected formation decay constant in accordancewith a function of said ratio and said initial formation decay constantto correct for any standoff from the borehole wall encountered by saiddetector as said logging tool traverses said borehole, and j) secondfunction former means for producing a log of porosity versus depthwithin said formation corrected for detector standoff from the boreholewall as a function of said standoff corrected formation decay constantderived from measurements in borehole models at known porosities andconditions of detector standoff from the borehole wall.
 9. The apparatusof claim 8 wherein said porosity log is standoff corrected for anyborehole enlargement that occurs in juxtaposition with said at least oneradiation detector assembly as said logging tool traverses along saidborehole.
 10. The apparatus of claim 8 wherein said porosity log isstandoff corrected for pulling away of the at least one radiationdetector assembly from contact with the surface of the borehole wall assaid logging tool traverses along said borehole.